PHY.K02UF Molecular and Solid State Physics

Cubic equations

A cubic equation has the form,

\begin{equation} ax^3 + bx^2 + cx + d = 0. \end{equation}

The determinant of this equation is defined by,

\begin{equation} \Delta = 18abcd - 4b^3d + b^2c^2 - 4ac^3 - 27a^2d^2. \end{equation}

If $\Delta > 0$, there are three distinct real roots. If $\Delta =0$, there is a multiple root and all roots are real. If $\Delta < 0$, there is one real root and two nonreal complex conjugate roots.

The critical points are where the slope is zero. These can be found by setting the derivative equal to zero.

\begin{equation} 3ax^2 + 2bx + c = 0. \end{equation}

If $b^2 -3ac > 0$, the critical points are located at,

\begin{equation} x_{\text{crit}}= \frac{-b\pm\sqrt{b^2-3ac}}{3a}, \end{equation}

and there is one local maximum and one local minimum. If $b^2 -3ac = 0$, there is one critical point at $x_{\text{crit}}=-\frac{b}{3a}$. This critical point is an inflection point. If $b^2 -3ac < 0$, there are no critical points.

The form below finds the roots of a cubic equation.

$a = $
$b = $
$c = $
$d = $